Naruto starts with the misadventures of a bratty outcast in search for recognition, and ends with saving the planet from an Assimilation Plot.
Neon Genesis Evangelion plays with the trope: every fight has the fate of humanity hang in the balance, but it's all a plan by a few fractions of humanity to initiate the end of the world on their own terms. As their plans collide, the end of the world ends up in the hands of a mentally fractured teenager, with disastrous results.
Code Geass starts being about a fallen prince's vendetta against his country of origin and a terrorist group struggling for their country's independence, and ends up being about saving the world from one Assimilation Plot and two nigh-omnicidal well-intentioned extremists in quick succession.
Part 6 of Jojos Bizarre Adventure follows this. It starts as a prison drama about an inmate trying to reclaim her father's memories, then becomes a race to prevent a crazy priest from resetting the universe. They fail.
Star Wars: Luke Skywalker is your average moisture farmer living in the dusty ass-end of nowhere when BAM! This random astromech droid starts playing back a message from a princess in need of saving from the Big Bad. Being the impressionable young man he is, Luke embarks on a mission to rescue the Damsel in Distress and maybe discover something about himself aling the way. Simple, right? Wrong. Before Luke has time to say 'I've got a bad feeling about this', he's undertaking a pivotal role in the war to overthrow the evil Empire and confront his destiny to decide the final battle between Good and Evil.
Oh, and before I forget, the damsel in distress that Luke totally open-mouth-tongue kisses is actually his twin sister and the Big Bad is his biological father.
Constantine: John Constantine is hired to investigate a suicide. In the end, he tries to stop a conspiracy to bring the son of the Devil on Earth.
TRON: Legacy is originally about rescuing the previous film's protagonist from imprisonment in virtual reality, and ends up being about saving the world from an invasion by a fascist AI.
2013 comedy The World's End, the last film of the "Three Flavors Cornetto" trilogy is about a group of old friends who reunite to re-enact a legendary pub crawl from their youth. At some point their pub crawl becomes a battle to save the human race.
China Miéville's Kraken starts with Billy Harrow investigating a minor mystery about a disappearing squid, and eventually escalates to doomsday cults and eldritch horrors trying to end the world.
Babylon 5 starts out with rather minor border skirmishes between the Centauri and Narn, some diplomatic tensions among other races, and an Earth Government moving slowly in a xenophobic direction. When theShadows reemerge into galactic affairs, they manipulate and cultivate these existing tensions into a galaxy-wide war among the younger races which escalates to an apocalyptic scale.
Star Trek: Deep Space Nine starts with the Federation taking over an old starbase from the Cardassians who'd recently withdrawn from a long brutal occupation of the planet Bajor which the starbase orbits. Episodes involve the rebuilding of Bajor and its various growing pangs of independence (and bitterness over its recent past), and some exploration through a wormhole recently discovered near the station. But a great power lay on the other side of that wormhole, which soon puts the whole Alpha Quadrant in jeopardy in the large-scale Dominion War.
Buffy the Vampire Slayer: The titular character starts out just trying to kill the local vampires, but quickly starts having to thwart the attempts of an old, powerful vampire, the Master, who is trying to get out of the magical cage he is trapped in. By the end of the season, she has to stop the Master from opening the Hellmouth, and act which would lead to a flood of demons entering the world and destroying all human life. The second season escalates the threat from "just" loss of all human life to loss of the very existence of the world.
Supernatural: The show starts with two brothers hunting ghosts, werewolves and other supernatural creatures in the US while looking for their father. In the fifth season, they're trying to prevent the Biblical Apocalypse from occurring.
Final Fantasy IV begins with Cecil's demotion for questioning orders and being sent to deliver a package. This spirals out of control until you find yourself going to the Moon, fighting a Humongous Mecha that is going to annihilate all life on the planet, and then doing battle with the incarnation of all evil.
Final Fantasy VI starts off with a mysterious girl named Terra working for the empire due to a form of mind control. She has no knowledge of who or what she is. When she gets knocked out in a mission, a treasure hunter named Locke quickly helps protect her. Before long she's helping The Resistance fighting against The Empire. The Empire itself poses a threat to the world, but the emperor himself would never go so far as to destroy it, which his seemingly comic-relief jester Kefka goes ahead and does just that. The world now in ruins, the heroes know that they at least have to stop Kefka from destroying all existence since he'd already gone that far.
Final Fantasy VII starts with a resistance group known as Avalanche fighting against the Shinra Mega Corp. just to help improve the lives of the citizens that live in the slums. One thing turns to another and eventually they're fighting a psychotic Super Soldier who wants to destroy the world.
Xeno Gears starts off with Fei just living his normal life in a quiet village, devoid of conflict. Then The Empire invades and razes the town to the ground, forcing him to defend himself in the heat of the moment. One thing leads to another and then he's fighting against a man-made God to save the world.
The original campaign begins with the effort to find a cure for the plague ravaging Neverwinter. It ends with Neverwinter fighting a full-scale war, and you trying to prevent the entire Sword Coast from falling back under the dominion of a 30,000-year-old sarrukh queen.
Hordes of the Underdark begins with drow raids on Waterdeep, and ends with a goddamn archdevil trying to take over Faerun.
It's easy to forget these days that Mass Effect's Commander Shepard wasn't born killing Reapers. Minus out-of-character knowledge you start the first game expecting an apprenticeship of sorts under a turian Spectre named Nihlus. That isn't the case for long. And until the Reapers actually arrive in Mass Effect 3 the only folks who believe you and are willing to help are the ones who either know you personally (your squad and crew, mainly), and crazies like the Illusive Man.
Particularly galling in Illusion of Gaia, where your goal is generally exploring ancient ruins and collecting mystic statues.
Dragon Age: Origins plays with this trope. While technically you are saving the world by stopping the Blight (i.e. a vast horde of evilmonsters led by a corrupted Dragon-God), the game's codex makes it quite clear that failure on your part will not actually lead to the end of the world. Blights reoccur every few centuries in Thedas, so people who dedicate their lives to stopping them have created a military organization, the Grey Wardens, just for that purpose. If you do not succeed, then one of the other members of your organization, which is thousands strong, would finish it in your place. By stopping the Blight, all you really do is keep the country that you live in from being destroyed before the other Wardens could act. Your victory simply means that the threat ended before the rest of the world noticed the problem.
Dragon Age II averts this: Templars and mages will end up fighting one another, but how thatcomes about is up to Hawke, who can end up either a symbol of rebellion or an accessory to a war crime.
Warcraft III starts with Thrall leaving the continent to find another one where his people can find a place to leave in peace, while Arthas investigates an epidemic. He then fights against a growing army of undead that threatens his kingdom. The climax has every faction of the world making a Last Stand against The Legions of Hell who want to destroy all life in the universe.
Burning Rangers at first starts off as you playing as a futuristic fire fighting crew that tackles raging fires and rescue survivors. However, near the end of the game the Rangers discover an inferno satellite that is collecting space debris from across the galaxy that's on a crash course with Earth. The Rangers therefore quickly head into the ship's central core in an attempt to destroy the satellite and save the world.
The Order of the Stick: The first arc of the story concerns the Order carrying out a standard dungeon crawl in order to defeat a lich. Then the next arc reveals that within the dungeon was a gate to a prison dimension where a world-destroying Eldritch Abomination is kept, and the lich was planning to use the gate for nefarious ends. The rest of the story concerns the Order's efforts to keep the other gates from being used for evil, so that the abomination isn't set free to destroy the world.
City Face parodies this. It starts off with a pigeon named City Face attempting and failing to woo an attractive female pigeon. A fairy shows up to help—and she tells him that the continued existence of the world depends on City Face successfully winning the female pigeon's heart.
Worm starts out as a superhero story with primarily street-level conflicts. The last few arcs of the story, however, feature a threat that could wipe out humanity in not just one world, but every possible reality.